Our tile flooring cracked and I am wondering what the best method is to repair it without actually repairing the whole tile?
It would be easier for people to see what's going on if you move the rug to photograph the whole affected area. It's impossible to tell the extent or location of the crack.
Do you have a spare tile, or can you buy one? Do you know the cause of the crack (like something being dropped on it)?
With better information, we may be able to give you a better answer, but with no other information, I would guess that either the tile wasn't properly supported, or it was installed on a surface that flexes. Without fixing the cause, there is no real repair that will last.
Once the support surface is solid, the only repair that will look good will be replacing the tile, which is also the easiest and fastest solution.
If you can't find a matching tile, an artist may be able to mask the crack by painting filler to match the tile's pattern. That could be an improvement, but it will still be visible and won't hold up to wear.
lift the carpet and glue the chips back in with super glue then fill the void under the tile with two part builders filler. fill under the edges of the other tiles too.
That stuff has a yucky chemical smell so you'll probably want a window open or a ventilator running for a few days.
Once the filler has set put the carpet back down, you can rent knee kicker to help stretch it back into position.
You might consider clear silicone caulking (usually comes in a red tube and is commonly used to seal bathtub enclosures). What’s nice about it is that you can squeeze it right in there, it cleans up with a damp cloth after application, and dries in 24-48 hours. The clear color means it won’t detract from your flooring. Cheap, easy, fast, effective, not ugly.